Table Of ContentH.S. Kasana . K.D. Kumar
Introductory Operations Research
Harvir Singh Kasana . Krishna Dev Kumar
Introductory
Operations Research
Theory and Applications
With 58 Figures
~ Springer
Prof. Dr. Harvir Singh Kasana
Thapar Institute Engineering and Technology
Department of Mathematics
147001 Patiala
India
Dr. Krishna Dev Kumar
Korea Advanced Institute of Science and Technology
Department of Aerospace Engineering
Kusong-dong 373-1, Yusong-gu
305-701 Taejon
Republic of Korea
[email protected]
Library of Congress Control Number: 2004106672
ISBN 978-3-642-07270-3 ISBN 978-3-662-08011-5 (eBook)
DOI 10.1007/978-3-662-08011-5
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Preface
This book is an outcome of the lectures delivered by the authors for
engineering and management students at the Birla Institute of Tech
nology and Science, Pilani, India. However, the text started when the
author shifted to the Thapar Institute of Engineering and Technology,
Patiala and the coauthor shifted to the Indian Institute of Technology,
Kanpur. During the teaching of this course, the authors realized a
need of a good text on "Optimization Research" and its applications
which may give comprehensive idea of various concepts and can be
used as a companion for problem solving techniques.
The primary purpose of this text is to bring this new mathematical
formalism into the education system, not merely for its own sake, but
as a basic framework for characterizing the full scope of the concept
of modern approach. The authors have tried all contents of this book
utilizing four hours a week in one semester as a core course. The level
of this book assumes that the reader is well acquainted with elementary
calculus and linear algebra. Being a textbook, we have taken enough
care so that the reader may attempt different type of problems.
Anyone who aspires to some managerial assignment or who is
a part of decision making body will find an understanding of opti
mization techniques very useful. The book is applied in orientation
with a concentration on engineering and management problems. Each
concept has been discussed with sufficient mathematical background
supported by examples. A set of problems has been added in the end
of every chapter.
Because of the imposed restriction of writing a relatively brief text
book on an extensive subject area, a number of choices had to be made
relating to the inclusion and exclusion of certain topics. No obvious
way of resolving this problem exists. The basic thinking is centralized
about the theme how the reader may continue to read advanced level
VI PREFACE
textbooks by self-study to develop research oriented thoughts. Hence,
the fundamental techniques have been emphasized while highly spe
cialized topics should be relegated to secondary one.
During the course of writing this book we have received remarkable
encouragement from our esteemed colleagues Prof. S. R. Yadava, and
Dr. S. P. Yadava at BITS, Pilani, India.
H. S. Kasana
K. D. Kumar
Contents
Preface v
1 Formulation 1
1.1 The Scope of Optimization 1
1.2 Introduction...... 4
1.3 Formulation of Models 9
2 Geometry of Linear Programming 33
2.1 Geometric Interpretation. . . . . . . . . . . . 33
2.2 Extreme Points and Basic Feasible Solutions. 45
2.3 Fundamental Theorem of Linear Programming 55
2.4 Graphical Method . . . . . . . . . . . . . . . . 56
3 The Simplex Algorithm 63
3.1 Introduction...... 63
3.2 The Simplex Algorithm 64
3.3 The Big-M Method. 80
3.4 Two Phase Method . 83
3.5 Exceptional Cases in LPP 89
4 Duality Theory 107
4.1 Dual Linear Program. . 107
4.2 Duality Theorems .. . 112
VIII CONTENTS
4.3 Complementary Slackness Theorem. . . · 114
4.4 An Economic Interpretation of Duality . · 118
4.5 The Dual Simplex Method. . . . . . . . · 119
5 Advanced Linear Programming 131
5.1 The Revised Simplex Algorithm. · 131
5.2 Complexity of The Simplex Algorithm · 143
5.3 Bounded Variable Technique · 144
5.4 Decomposition Principle ... · 151
5.5 Karmarkar Interior Point Algorithm. · 156
6 Sensitivity Analysis 169
6.1 Introduction ........ . · 169
6.2 Change in the Cost Vector. · 170
6.3 Changes in the Right-hand Side Vector. · 175
6.4 Change in the Constraint Matrix · 179
6.5 Special Cases . . . . . . . · 190
6.6 Parametric Programming · 192
7 Transportation Problems 205
7.1 Introduction ...... . .205
7.2 Optimal Solution from BFS .213
7.3 Unbalanced Transportation Problem · 221
7.4 Transshipment .... .226
7.5 Assignment Problems .232
7.6 Hungarian Method .234
8 Network Analysis 253
8.1 Introduction to Networks ..... .253
8.2 Minimal Spanning Tree Algorithm .254
8.3 Shortest Path Problem. .259
8.4 Maximal Flow Problem .267
CONTENTS IX
9 Project Management 277
9.1 Introduction. . .. .277
9.2 Critical Path Method .278
9.3 Critical Path Determination. . 281
9.4 Optimal Scheduling by CPM .283
9.5 Project Evaluation and Review Technique .293
10 Sequencing Problems 307
10.1 Introduction. . . . . . . . . . . . . . 307
10.2 Problem of n Jobs and 2 Machines . 308
10.3 Problem of n Jobs and m Machines. . 313
10.4 Two Jobs on Ordered m Machines . 317
11 Integer Programming 325
11.1 Introduction. . . . . . . . . . . 325
11.2 Branch and Bound Algorithm . 326
11.3 Traveling Salesman Problem. . 331
11.4 Cargo Loading Problem . 340
12 Dynamic Programming 349
12.1 Introduction. . 349
12.2 Formulation. . 350
12.3 Recursive Relations. . 355
12.4 Continuous Cases. . 358
12.5 Discrete Cases . 361
12.6 Forward Recursions. . 369
12.7 Linear Programming vs Dynamic Programming . 370
13 Nonlinear Programming 377
13.1 Introduction. . . . . . . 377
13.2 Lagrange Multipliers Method . 381
13.3 Convex Nonlinear Programming Problem . 386
x CONTENTS
13.4 Kuhn Tucker Theory .. .390
13.5 Quadratic Programming .393
13.6 Separable Programming .398
13.7 Duality in Nonlinear Programming .405
14 Search Techniques 413
14.1 Unimodal function ..... . .413
14.2 Dichotomous Search Method .415
14.3 Fibonacci Search Method .417
14.4 Golden Section Method . .421
14.5 Steepest Descent Method .422
14.6 Conjugate Gradient Method. .425
15 Geometric Programming 433
15.1 Introduction ................ . .433
15.2 Unconstrained Posynomial Optimization. .438
15.3 Constrained Posynomial Optimization .446
16 Goal Programming 457
16.1 Introduction ... .457
16.2 Standard form of LGPP .459
16.3 Partitioning Algorithm . .460
16.4 Grouping Algorithm .468
17 Games Theory 495
17.1 Introduction ................... . .495
17.2 Two Person Zero Sum Game (Pure Strategies) .496
17.3 Two Person Zero Sum Game (Mixed Strategies) .499
17.4 Games Theory vs Linear Programming. .501
18 Special Topics 509
18.1 Extremum Difference Method .509
CONTENTS XI
18.2 Generalized Transportation Problem .511
18.3 Generalized Assignment Problem .. .517
18.4 Multiobjective Transportation Problem .520
Appendix: Objective Type Questions 529
Bibliography 549
Answers 553
Index 577
